First, you have the LEDs on the wrong side, they should be on the TTL side because they unnecesarily load the charge pump voltage.
Second, one is connected to input, the other to output, and both these channels are left open at the TTL side, so no wonder nothing happens.

Are you sure you understand how the chip works? T1IN changes T1OUT, R1IN changes R1OUT and so forth, T2 and R2 are completely independent of T1 and R1.

Then try connecting them the other way, not connected to ground but connected to +5V. You could also use a tad higher values of the led resistors so that you don´t load the ttl outputs too much, 5mA should be enough to see.

Serial port idle is high level on ttl side and low level on rs232 side as you confirmed with scope, so you need to invert the leds.

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I chose to use the transistor method so I can have a 3V or 5V MCU and not have to worry about changing the current limiting resistor though thinking it again I probably could have picked a happy medium for the resistor and do away with the transistors.

Plus I read the MAx data sheet only after I had the thing built and found it would operate at a +3v supply too.